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Theorem oteqex2 4274
Description: Equivalence of existence implied by equality of ordered triples. (Contributed by NM, 26-Apr-2015.)
Assertion
Ref Expression
oteqex2  |-  ( <. <. A ,  B >. ,  C >.  =  <. <. R ,  S >. ,  T >.  ->  ( C  e.  _V  <->  T  e.  _V ) )

Proof of Theorem oteqex2
StepHypRef Expression
1 opeqex 4273 . 2  |-  ( <. <. A ,  B >. ,  C >.  =  <. <. R ,  S >. ,  T >.  ->  ( (
<. A ,  B >.  e. 
_V  /\  C  e.  _V )  <->  ( <. R ,  S >.  e.  _V  /\  T  e.  _V )
) )
2 opex 4253 . . 3  |-  <. A ,  B >.  e.  _V
32biantrur 492 . 2  |-  ( C  e.  _V  <->  ( <. A ,  B >.  e.  _V  /\  C  e.  _V )
)
4 opex 4253 . . 3  |-  <. R ,  S >.  e.  _V
54biantrur 492 . 2  |-  ( T  e.  _V  <->  ( <. R ,  S >.  e.  _V  /\  T  e.  _V )
)
61, 3, 53bitr4g 279 1  |-  ( <. <. A ,  B >. ,  C >.  =  <. <. R ,  S >. ,  T >.  ->  ( C  e.  _V  <->  T  e.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656
This theorem is referenced by:  oteqex  4275
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662
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